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In how many ways can 10 engineers & 4 doctors be seated togther at a round table if all the 4 doctors sit together?
Since all the 4 doctors sit together then we consider them as a single person.
Total no. of person
11 persons seated at a round table in (111)! = 10! Ways.
4 doctors seated among themselves in 4! Ways.
Required no. of ways
Kiran has 8 blacks balls & 8 white balls. In how many ways can he arrange these balls in a row so that balls of different colours alternate?
8 white balls can be arranged in 8! Ways.
8 black balls can be arranged in 8! Ways.
If they alternate their place then it will happen
Required no.of ways
(Since we have to place one type & alternate another type ball)
How many factors of \({2^5}\times{3^6}\times{5^2}\) are perfect squares?
Any factor of which is a perfect square will be of the form where a can be 0 or 2 or 4 (3 ways) b can be 0 or 2 or 4 or 6 (4 ways) c can be 0 or 2 (2 ways)
Required no.of ways
Five balls needs to be placed in 3 boxes. Each box can be hold all the 5 balls. In how many ways can the balls be placed in the boxes if all the balls are identical and all boxes are different?
Hence n=3, k=5
Distribution of k balls into n boxes
In how many ways 7 different balls can be distributed in 5 different boxes if any box can contain any number of balls except that ball 3 can only be put into box 3 or box 4?
Ball 3 can only be put in box 3 or box 4 is 2 ways.
Other 6 balls can sequencialy put in 5 boxes is .
Total no. of ways
Five balls needs to be placed in 3 boxes. Each box can hold all the 5 balls. In how many ways can the balls be placed in the boxes if no box can be empty & all the boxes are different?
Hence n=3, k=5
Required no.of ways
In how many ways can 10 software engineers & 10 civil engineers be seated around a round table so that they are positioned alternatively?
10 civil engineers can be arranged around a round table in (101)!=9! Ways.
10 software engineers can be arranged in 10! Ways.
As they are positioned alternately then required no.of ways
A company has 10 civil engineers & 6 software engineers. In how many ways they can be seated around a round table so that all the software engineers are seated together?
Since all the software engineers are seated together we will take them one person.
Hence the total no. of persons =10+1=11
11 persons can be seated in a round table= (111)!=10! ways.
6 Software enginers arranged their seats between themselves is 6! ways.
Total no.of ways
How many 8 digit mobile no.s can be formed if any digit can be repeated and 0 can also start the mobile number?
All the 10 digit no.s can be repeated & 0 can be start the mobile number.
Hence the required no.of ways is
Five balls need to be palced in 3 boxes. Each box can hold all the five balls. In how many ways can the balls be placed in the boxes so that no box can be empty if all balls are different but all boxes are identical?
Hence n=3, k=5
Required no.of ways =s(k,n)
=
=
Five balls need to be placed in 3 boxes. Each box can hold all the five balls. In how many ways can the balls be placed in the boxes so that no box remains empty, if all balls and boxes are identical?
Hence n=3, k=5
P(k,n) = P(5,3)
The partition of 5 into 3 parts are
1+1+3
1+2+2
Therefore, no.of partition of 5 into 3 parts =2
How many signals can be made using 6 different coloured flags when any no. of them can be hoisted at a time?
No.of signals can be made using 1 flags, 2 flags, 3 flags, 4 flags, 5 flags, 6 flags are respectively.
Therefore, required no. of ways
=
=6+30+120+360+720+720
=1956
How many possible outcomes are there when 5 dice are rolled in which at last one dice shows 6?
No. of possible outcomes when 5 dice are rolled
No. of possible outcomes when 5 dice are rolled in which 6 does not appear in any dice
No. of possible outcomes when 5 dice are rolled in which at least one dice shows
A board meeting of a company is organized In a room for 24 persons along the two sides of a table with 12 chairs in each side. 6 persons wants to sit on a particular side & 3 persons wants to sit on the other side. In how many ways can they be seated?
6 Persons arranged in 12 chairs in ways.
3 Persons arranged in 12 chairs in ways.
Remaning Persons =2463=15
Ramaining chairs =2463=15
15 persons arranged in 15 chairs in 15! Ways.
Total no. of ways
How many numbers not exceeding 10000 can be made using the digits 2,4,5,6,8 if repetition of digits is allowed?
Count of 1digit no. is
Count of 2digit no., 3digit no., 4digit no. is
Hence total count
How many 5digit no.s can be formed using the digits 1,2,3,4,…,9 s.t. no two consecutive digits are same?
The place can be filled by 9 ways.
Since the consecutive digits are not same rest 4 places can be filled by 8, 8, 8, 8 ways.
Required no.s are
In how many ways can 5 blue balls. 4 white balls & the rest 6 different colours balls be arranged in a row?
Hence, all balls are not different
Total no. of balls=5+4+6=15
No. of blue balls =5
No. of white balls =4
Rest six balls are of different balls.
So we take all the different colour balls as a single ball.
Total no.of ways
A company has 10 software engineers and 6 mechanical engineers. In how many ways can a committee of 4 engineers be formed from them s.t. the committee must contain exactly 1 mechanical engineer?
No.of ways this can be done
From a group of 8 women & 6 men, a committee consisting of 3 men & 3 women is to be formed. In how many ways the committee to be formed?
3 mens choosen from 6 men in ways.
3 women choosen from 8 women in ways.
Total no. of ways
How many 2digit no.s can be formed by using the digits 0,2,4,7,9, if repetition is not allowed?
Total digit = 5
No.of ways is
=16
Since 0 can’t placed in place.